We first consider the black hole, where the tortoise coordinate is related to the standard radial coordinate \(r\)  of the Schwarzschild metric by

x=r+2M(r-2M2M)+(2(2)-3)M+xlr
.
We have added some constants and the significance of \(x_{\rm lr}\)  will become clear shortly. The range \(-\infty <x<\infty \)  corresponds to \(2M<r<\infty \) . For an observer at infinity, \(\Delta x\)  gives the light travel time to cover a radial distance \(\Delta r\) . The function \(R(x)\)  for a black hole is obtained by inverting () for \(r=R_{\rm bh}(x)\) , which can be done explicitly.This was noticed for example in [1]} and it can be done in Maple. Thus the black hole is represented by the metric in () when

Rbh(x)=2M[1+W(12(x-xlr+M2M))],
